FM04 12:lO
Proceedings of the 37th IEEE
Conference on Decision & Control
Tampa, Florida USA December 1998
Optimal Robust Disturbance Attenuation for
Continuous Time-Varying Systems
Mohamed Seddik Djouadi
Centre for Intelligent Machines, and
Dept. of Electrical Eng., McGill University
3480 University Street
Montrkal, Qukbec, Canada
Corresp. address Dept. of Electrical Eng.
Wayne State University
5050 Anthony Wayne Drive # 3100
Detroit, Michigan, USA 48202
djouadi@ece.eng.wayne.edu, djouadi@cim.mcgill.ca
and for h < 0 by:
Abstract
Dh :
In this paper we consider the Optimal Robust
Disturbance Attenuation Problem (ORDAP) for
continuous time-varying systems subject to timevarying unstructured uncertainty. We show that
for causal (possibly time-varying) continuous systems, ORDAP is equivalent to finding the smallest fixed point of a “two-disk” type optimization
problem under time-varying feedback control laws.
Duality is applied in the context of nest algebra of causal stable systems, to prove existence of
optimal continuous time-varying controllers. We
also show that for time-invariant nominal plants,
time-varying control laws offer no improvement
over time-invariant feedback control laws and hence
generalizing previous results obtained for discrete
time-varying systems
Definitions and Notation
B ( E , F ) denote the space of bounded linear
operators from a Banach space E to a Banach
space F , endowed with the operator norm.
L”0, CO) the standard Lebesgue space of essentially square integrable functions defined
on the interval [O,m).
L2[0,00) --+ L2[[o,00)
f(t) --+f ( t - h )
0-7803-4394-8198
$10.000 1998 IEEE
(1)
(2)
At) -4 f(t - h)X[O,m)( t )
where X [ O , ~ is
) the characteristic function of
the interval [0,00).An operator A E ’B(E,F )
is said to be time-invariant if, for all h 2 0,
A satisfies the operator equation [l]
P, the usual truncation operator, which sets
all outputs after time T to zero.
0
An operator A E % ( E ,F ) is said to be causal
if it satisfies the operator equation:
P,AP, = PTA, Vr E (0,m)
(4)
The subscripts “sc”, ‘‘c” and the superscript ‘‘ti”
denote the restriction of a subspace of operators
to its intersection with causal, strictly causal (see
[2] for definition), and time-invariant operators respectively. * stands for the adjoint of an operator
or the dual space of a Banach space depending on
the context. -* denotes convergence in the weak*
topology.
Dh the family of delay operators defined for
h 2 0 by:
Dh :
L 2 [ 0 , ~-+
) L2[o,m)
1 Introduction
The Optimal Robust Disturbance Attenuation
Problem (ORDAP) was formulated by Zames 131,
and considered by Bird, Francis, Owen and Zames
381 9
If a particular controller Q achieves a "worst-case"
weighted sensitivity function less than some r > 0
then,
[4, 5, 6, 7, 81. In ORDAP a stable uncertain plant
P is subject to disturbances at the output. The
objective is t o find a feedback control law which
provides the best uniform attenuation of uncertain
output disturbances in spite of uncertainty in the
plant model. Here we examine the ORDAP for
possibly time-varying continuous systems and feedback control laws.
Analysis of time-varying control strategies for optimal disturbance rejection for known time-invariant
plants has been studied by Shamma and M.A.
Dahleh [9], Chapellat and M. Dahleh [l]. A robust version of these problems were considered
in [lo, 11, 121 in different induced norm topologies. They showed that for time-invariant nominal
plants, time-varying control laws offer no advantage over time-invariant ones.
The ORDAP for discrete timevarying systems and
feedback laws has been studied by Owen [7,13]. He
characterized the problem in a predual space and
proved that in some induced norm topology the
same conclusion as before holds. In this paper we
analyse the ORDAP for continuous time-varying
systems extending previous results and therefore
settling an open question in [7].
I\W(I - P o Q ) ( I + XVPoQ)-'I1
(I XVP,Q)-l E B c ( L 2 ,L'),
vx E IB,(L"LL'), llxll < 1
I r , and
+
(8)
Expression (8) is equivalent to:
Ilw(I- P o Q ) ( I + xVf'oQ)-'fllp
Lrllfll~z,
and ( I f XVP,Q)-l E 'Bc(L2,LL2),
(9)
vx E B,(L"L'), llXll < 1, Vf E L"0,oo)
yielding
I l W ( I - P 0 Q ) F l l p I rll(I + X V P o Q ) F I I ~ 2 and
,
( I XVPoQ)-l E B,(L2, L'), VX E !Bc(L2,L'),
+
IlXll < 1, V F E L"O,oo), IIFllU L 1
(10)
which is in turn implied by (see chapter 5 of [ 2 ] ) :
Ilw(1-PoQ)FlILz I r l l F l l ~ 2- r l l v p o Q F I I ~ ~ ,
V F E L 2 , llFll~z5 1
(11)
Hence:
2 Problem Formulation
Let Po E IBB,,(L2[0,
m),L2[0,m))be the nominal
(possibly time-varying) plant, and denote the set
of plant uncertainty by:
e ( P o , V ) = { P E IBc(L"-oo,m),L"-oo,oo))
:
P = XVP, Po,
(5)
E IB,(L"-m, m),L"-oo, m)),J J X<
JJ
1)
+
x
where V is a causal, linear time-invariant weighting
function.
The ORDAP can be shown t o be equivalent to
finding the optimal worst case sensitivity function
with respect to disturbances and plants in e(P0,V ) ,
achievable by a feedback control law. Mathematically this is equivalent to
Po =
inf
sup
(1
IIW(I+
C stabilizing PEC(PO,V)
p E e(po,v)
inf
Q E %(L2, L2)
( I + XVP,Q)-' E IB(L"LL")
((W(I PoQ)(I
+ XVPoQ)-lII
+ r IIVPoQfllLa )
then [ is a continuous, positive, non-decreasing
function of T . We get a similar theorem to theorem 6.1 for discrete time-varying systems in [7]:
Theorem 1 1) Let Po be time-invariant and po
as above, if there exists an optimal Q E 'Bc(L2,L 2 )
for each r E [0,11 in the expression (141, then po
is equal to the smallest f i e d point of [ ( r ).
2) If Po is time-varying then po is bounded above
b y the smallest fixed point of [ ( r ) .
Proof: similar t o the discrete time case (see [14]).
Hence the optimization (7) for time-invariant nominal plant reduces to:
(6)
where W is a causal, linear, time-invariant weighting function. Expression (6) can be written as:
Po =
(IlW(I - P o Q ) f l l u
SUP
PE C(P0 , V )
(7)
3820
Next we proceed t o give the duality structure of
the problem, which shows existence of an optimal
Q for P O .
Let lL2 be the Banach space L2 x L2 under the
norm:
3 Duality Structure and Existence of an
The following lemma characterizing the dual space
of C2 follows easily from [li']:
Optimal Solution
Let C2 be the Banach space L2[0,CO) x L2[0,00)
under the norm:
The vector function
(
Lemma 1
)
a
(L2)* N C2
a
(C2)* N IL2
w ( ~ ~ o can
~ Qbe )
Hence all these Banach spaces are reflexive.
viewed as an operator fr'om L2[0,co)into C2 with
the operator induced norm:
Define N(B1,B2) t o be the Banach space of nuclear operators mapping the Banach space B1 to
the Banach space B2 under the nuclear norm. Recall that an operator A : B1 -+ B2 is said to be
nuclear if it has the representation [MI:
Therefore, po can be expressed as a distance prob-
( y ) IB(L2,P)
= ( 7 ) Po B c ( L 2 , L 2 )
of
lem from the vector function
to the subspace S
n
+
where <
> is the duality product and the nuclear norm is defined to be:
a , .
+
( ) Al'i71,
IIAtInuc
E
IICII.
ItenIII
(21)
where the infimum is taken over all possible representations of A.
The trace of the nuclear operator A is denoted by
tr(A) and is defined by
assumption
!Bc(L2,L 2 )
=inf{C
n
(A) implies that the operator R*R E B ( L 2 , L 2 )
has a bounded inverse, this ensures closedness of
S According t o Arveson ([16], Corollary 2, see also
[2]), the self-adjoint operator R* R has a spectral
factorization of the form:
R"R = A"& where A , A-'
(20)
n
B ( L 2 , L 2 )We
) . assume that IW(jw)12 IV(jw)12>
0 , 0 5 w 5 00. Then there exists an outer spectral factor A1 E H m , invertible in Hm, such that
lA1(jw)l2 = IW(jw)12 lV(jw)12. Therefore Alpo
is a bounded linear operator in B c ( L 2 L
, 2 )has a polar decomposition UlG, where U1 is a partial isometry and G a positive operator both defined on L2
[15]. Next we assume (A) U1 is unitary and G f l E
IB(L2,L2). Let R =
llF,*Il . IlenII < 00
F,* E B,*,and
of
< F,*,Fn >
tr(A) =
n
This sum is well defined and can be shown to be independent of the representation [ 181. The following
lemma applies to the Banach spaces L2 and lL2.
Define
Lemma 2 [18, 11
R2 = RA-'
then R;R2 = I , and S has the equivalent representation, S = R2Bc(L2,L2)). After absorbing A
in the free parameter Q, the optimization problem
(14) is then equivalent to:
B(L2, C2) Y N ( P ,P)*
i f 4 E N ( L 2 , L 2 ) * the
, isometric isomorphism
given by:
$(A) =< B , A >= trA*B = trB*A,
where A E N ( L 2 , L 2 ) , B E B ( L 2 , f ? )
3821
(23)
2~
is
Next, we characterize the preannihilator of the
the Banach space of causal operators 'B,(L2,L 2 )
as a subspace of B ( L 2 , L 2 )in the space of nuclear operators N ( L 2 , L 2 ) . But first define for each
t E (-00, CO), Pt ( L 2 )the subspace of L2 consisting of all functions f in L2 such that f = 0, a.e.
on ( t , 00). Then A4 = { P t ( L z ) , -00 < t < 0 0 ) is
a continuous nest [15]. The space !J3,(L2,L 2 ) may
then be viewed as a nest algebra since causal operators leave M invariant, i.e., AMt C Mt, for all
Mt E M .
The Banach space N ( L 2 , L 2 ) is nothing but the
well-known space of trace-class operators from L2
t o L 2 , and that the nuclear norm in this case reduces t o the trace-norm [9]. Applying theorem 16.6
[15], we get the following lemma.
Lemma 3 N c ( L 2 , L 2 ) is the preannihilator of
23@,
L").
Proof
the
preannihilator
of
Using a standard result from Banach space duality theory relating the distance from a vector to a
subspace and an extremal functional in the predual
(see [19]), we deduce the following theorem.
Theorem 2 Under assumption ( A ) there exists at
least one Q o E 'B,(L',L2), i.e., a linear timevarying control law such that:
A E IS
23,(L2,L'),
4 Time-Varying versus Time-Invariant
' B , ( L 2 , L 2 ) is defined by
Control laws
''Bc(L2,L 2 ) = {A E N ( L 2 , L 2 ) : t r ( T A ) = 0 ,
VT E 'B,(L2,L')}
(24)
Let M; be the closed linear span of { M , E M
: 7 < t } , but since M is a continuous nest, then
M; = Mt. It follows by theorem 16.6 [15] that
' S , ( L 2 ,L 2 ) = N c ( L 2 , L').
Now let S, be the subspace of N(C2,L2(0,cm)))
defined as follows:
S, =
{ p ( Q ( I - R2R;)
+ AR;) (La :
@ E N(,CE,L'),
A
In the next theorem, we show that no advantage
is gained in performance if we allow our controllers
to be time-varying, and thus generalizing the same
result obtained for discrete time systems in [7, 131.
(25)
E N c ( L 2 ,L')}
where p is the canonical projection of L2(--oo,00)
into L 2 [ 0 , m ) ,and CL is defined t o be the Banach
space L 2 ( - m , 00) x L ' ( - ~ o ,00) under the norm of
,!?. Define the following subspace of N ( L 2 , L 2 ) :
IS = { A E N(L2,1L2): A*
E S,}
Lemma 4 IS is the preannihilator of S
In this section, we assume that the nominal plant
Po is time-invariant. It follows also that Rz is also
time-invariant. Define the following performance
index when the controllers are constrained to be
linear time-invariant:
Theorem 3 If Po k linear, causal, time invariant
and assumption ( A ) holds, then
P o = P:
(26)
(29)
Proof since 23p (L" L 2 ) c 'Bc(L" L 2 ) ,it is obvious
that p o 5 p z . Let Q E 'Bc(L2,L2)
in
N (L 2 ,L2).
Proof let T E 'B(L2(O,0o),f?), then
<T,A>=O, V A E ' S ~
such a Q exist by theorem 2. Using an idea of
Chapellat and M. Dahleh [l],let h > 0. Since
llD,(( 5 1, Vu, we have that Vn > 0
+
t r ( p ( @ ( I - RzRI;)(,,T pNR;I,z 7') = 0
V* E N ( C 5 , L 2 [ O , o o ) ) , NE N c ( L 2 , L 2 )
U ( I - R2R;)IpTp = 0
and R 2 R ; l p T p E B C ( L 2L, 2 ) by lemma 3
3822
For all h
Using time-invariance of W and R2, we get
Define for all n 2 0
.
n
then (32) yields
IlQn(h)Il 5 11Q11, vn 2 0 and then llR2Qn(h)ll is
uniformly bounded by 11R21111Q11. Moreover the predual space N(L2,1L2)of B ( L 2 , L 2 ) ,by Alaoglu’s
theorem there exist a subsequence of {R2Qnk( h ) }
which converges in the weak* topology to some vector function R E B ( L 2 ,C 2 ) ,we write
by (34) and a property of weak* limits, we have
By definition of the weak* limit, V A E’ S , 0 =<
R 2 Q n k ( h ) , A>+< R , A >, as k + 00, hence R =
R z Q ( h ) , @ ( h ) E IB,(L2,LL”).The adjoint of Dh,
D i is defined by Vf E L2
( D f i f ) ( t=
) f ( t-k h ) = ( D - h f ) ( t )
Now V A E N (L” lL2 ), we have
(37)
> 0, we showed that
Qn 2 1, define the sequence R2Qn = R2Q(:).
By-(39) it is bounded, therefore there 3Q s.t.
R2Qnk -* R2Q and R2Q satifies (40) as before. Again using properties of nuclear operators
it is easy to see that R2QnkDh - DhR2Qnk -*
R2QDh - R2Q. Using exactly the same procedure
as in [I] to prove that R2QnkDh - DhR2Qnk -* 0
shows that R2Q is time-invariant, but since R2 is
time-invariant, Q must also be time-invariant and
this completes the proof.
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